"The introduction of numbers as coordinates is an act of violence."
—Hermann Weyl
The fundamental, root-of-it-all building block for linear algebra is the vector, so it's
worth
making sure that we're all on the same page about what exactly a vector is.
You see, broadly
speaking there are three distinct but related ideas about vectors, which I'll call the physics
student perspective, the computer science student perspective, and the mathematician's
perspective.
The physics student perspective is that vectors are arrows pointing in space.
What defines a given
vector is its length, and the direction it's pointing in, but as long as those two facts
are the
same, you can move it all around and it's still the same vector.
Vectors that live in the flat plane
are two-dimensional, and those sitting in broader space that you and I live in are three-dimensional.
The computer science perspective is that vectors are ordered lists of numbers.
For example, let's
say that you were doing some analytics about house prices, and the only features you cared
about
were square footage and price.
You might model each house with a pair of numbers: the first
indicating square footage, and the second indicating price.
Notice that the order matters here.
In the lingo, you'd be modelling houses as two-dimensional vectors, where in this context,
"vector" is pretty much just a fancy word for "list", and what makes it two-dimensional
is the fact
that the length of that list is 2.
The mathematician, on the other hand, seeks to generalise both of these views, basically
saying that
a vector can be anything where there's a sensible notion of adding two vectors, and multiplying
a
vector by a number, operations that I'll talk about later on in this video.
The details of this view
are rather abstract, and I actually think it's healthy to ignore it until the last video
of this
series, favoring a more concrete setting in the interim,
but the reason that I bring it up here is that it hints at the fact that ideas of vector
addition
and multiplication by numbers will play an important role throughout linear algebra.
But before I talk about those operations, let's just settle in on a specific thought
to have in mind
when I say the word "vector".
Given the geometric focus that I'm shooting for here, whenever I
introduce a new topic involving vectors, I want you to first think about an arrow—and
specifically,
think about that arrow inside a coordinate system, like the x-y plane, with its tail
sitting at the origin.
This is a little bit different from the physics student perspective, where vectors can freely
sit
anywhere they want in space.
In linear algebra, it's almost always the case that your vector will be
rooted at the origin.
Then, once you understand a new concept in the context of arrows in space,
we'll translate it over to the list-of-numbers point-of-view, which we can do by considering
the coordinates of the vector.
Now while I'm sure that many of you are familiar with this coordinate system, it's worth walking
through explicitly, since this is where all of the important back-and-forth happens between
the two
perspectives of linear algebra.
Focusing our attention on two dimensions for the moment, you have a
horizontal line, called the x-axis, and a vertical line, called the y-axis.
The place where they
intersect is called the origin, which you should think of as the center of space and
the root of all vectors.
After choosing an arbitrary length to represent 1, you make tick-marks on each axis to
represent this distance.
When I want to convey the idea of 2-D space as a whole, which you'll see
comes up a lot in these videos, I'll extend these tick-marks to make grid-lines, but right
now
they'll actually get a little bit in the way.
The coordinates of a vector is a pair of numbers that
basically give instructions for how to get from the tail of that vector—at the origin—to
its tip.
The first number tells you how far to walk along the x-axis—positive numbers indicating
rightward
motion, negative numbers indicating leftward motion—and the second number tell you how
far to walk
parallel to the y-axis after that—positive numbers indicating upward motion, and negative
numbers
indicating downward motion.
To distinguish vectors from points, the convention is to write this pair
of numbers vertically with square brackets around them.
Every pair of numbers gives you one and only one vector, and every vector is associated
with one and
only one pair of numbers.
What about in three dimensions?
Well, you add a third axis, called the z-axis,
which is perpendicular to both the x- and y-axes, and in this case each vector is associated
with an ordered triplet of numbers: the first tells you how far to move along the x-axis,
the second
tells you how far to move parallel to the y-axis, and the third one tells you how far
to then move
parallel to this new z-axis.
Every triplet of numbers gives you one unique vector in space, and
every vector in space gives you exactly one triplet of numbers.
So back to vector addition, and multiplication by numbers.
After all, every topic in linear algebra
is going to center around these two operations.
Luckily, each one is pretty straightforward to define.
Let's say we have two vectors, one pointing up, and a little to the right, and the other
one
pointing right, and down a bit.
To add these two vectors, move the second one so that its tail sits
at the tip of the first one; then if you draw a new vector from the tail of the first one
to where
the tip of the second one now sits, that new vector is their sum.
This definition of addition, by the way, is pretty much the only time in linear algebra
where we let
vectors stray away from the origin.
Now why is this a reasonable thing to do?—Why this definition of addition and not some other
one?
Well the way I like to think about it is that each vector represents a certain movement—a
step with
a certain distance and direction in space.
If you take a step along the first vector,
then take a step in the direction and distance described by the second vector, the overall
effect is
just the same as if you moved along the sum of those two vectors to start with.
You could think about this as an extension of how we think about adding numbers on a
number line.
One way that we teach kids to think about this, say with 2+5, is to think of moving
2 steps to the
right, followed by another 5 steps to the right.
The overall effect is the same as if you just took
7 steps to the right.
In fact, let's see how vector addition looks numerically.
The first vector
here has coordinates (1,2), and the second one has coordinates (3,-1).
When you take the vector sum
using this tip-to-tail method, you can think of a four-step path from the origin to the
tip of the
second vector: "walk 1 to the right, then 2 up, then 3 to the right, then 1 down."
Re-organising
these steps so that you first do all of the rightward motion, then do all of the vertical
motion,
you can read it as saying, "first move 1+3 to the right, then move 2+(-1) up," so the
new vector has
coordinates 1+3 and 2+(-1).
In general, vector addition in this list-of-numbers conception looks
like matching up their terms, and adding each one together.
The other fundamental vector operation is multiplication by a number.
Now this is best understood
just by looking at a few examples.
If you take the number 2, and multiply it by a given vector, it
means you stretch out that vector so that it's 2 times as long as when you started.
If you multiply
that vector by, say, 1/3, it means you squish it down so that it's 1/3 of the original length.
When you multiply it by a negative number, like -1.8, then the vector first gets flipped
around,
then stretched out by that factor of 1.8.
This process of stretching or squishing or sometimes reversing the direction of a vector
is called "scaling",
and whenever you catch a number like 2 or 1/3 or -1.8 acting like this—scaling some
vector—you call it a "scalar".
In fact, throughout linear algebra, one of the main things that
numbers do is scale vectors, so it's common to use the word "scalar" pretty much interchangeably
with the word "number".
Numerically, stretching out a vector by a factor of, say, 2, corresponds to
multiplying each of its components by that factor, 2, so in the conception of vectors
as
lists of numbers, multiplying a given vector by a scalar means multiplying each one of
those components by that scalar.
You'll see in the following videos what I mean when I say that linear algebra topics
tend to revolve
around these two fundamental operations: vector addition, and scalar multiplication; and I'll
talk
more in the last video about how and why the mathematician thinks only about these operations,
independent and abstracted away from however you choose to represent vectors.
In truth, it doesn't
matter whether you think about vectors as fundamentally being arrows in space—like
I'm suggesting
you do—that happen to have a nice numerical representation, or fundamentally as lists
of numbers
that happen to have a nice geometric interpretation.
The usefulness of linear algebra has less to do with
either one of these views than it does with the ability to translate back and forth between
them.
It gives the data analyst a nice way to conceptualise many lists of numbers in a visual way,
which can seriously clarify patterns in data, and give a global view of what certain operations
do,
and on the flip side, it gives people like physicists and computer graphics programmers
a language
to describe space and the manipulation of space using numbers that can be crunched and
run through a computer.
When I do math-y animations, for example, I start by thinking about what's actually
going on in
space, and then get the computer to represent things numerically, thereby figuring out where
to
place the pixels on the screen, and doing that usually relies on a lot of linear algebra
understanding.
So there are your vector basics, and in the next video I'll start getting into some pretty
neat
concepts surrounding vectors, like span, bases, and linear dependence.
See you then!
Captioned by Navjivan Pal